In 1993, Baxter gave eigenvalues of the transfer matrix of the N -state superintegrable chiral Potts model with the spin-translation quantum number Q , where m Q = ⌊( NL − L − Q )/ N ⌋. In our previous paper we studied the Q = 0 ground-state sector, when the size L of the transfer matrix is chosen to be a multiple of N . It was shown that the corresponding τ 2 matrix has a degenerate eigenspace generated by the generators of r = m 0 simple algebras. These results enable us to express the transfer matrix in the subspace in terms of these generators E ± m and H m for m = 1, ..., r . Moreover, the corresponding 2 r eigenvectors of the transfer matrix are expressed in terms of rotated eigenvectors of H m .
It has been shown in earlier works that for Q = 0 and L a multiple of N , the ground state sector eigenspace of the superintegrable τ 2 ( t q ) model is highly degenerate and is generated by a quantum loop algebra . Furthermore, this loop algebra can be decomposed into r = ( N − 1) L / N simple algebras. For Q ≠ 0, we shall show here that the corresponding eigenspace of τ 2 ( t q ) is still highly degenerate, but splits into two spaces, each containing 2 r − 1 independent eigenvectors. The generators for the subalgebras, and also for the quantum loop subalgebra, are given generalizing those in the Q = 0 case. However, the Serre relations for the generators of the loop subalgebra are only proven for some states, tested on small systems and conjectured otherwise. Assuming their validity we construct the eigenvectors of the Q ≠ 0 ground state sectors for the transfer matrix of the superintegrable chiral Potts model.
In order to calculate correlation functions of the chiral Potts model, one only needs to study the eigenvectors of the superintegrable model. Here we start this study by looking for eigenvectors of the transfer matrix of the periodic τ 2 ( t q ) model which commutes with the chiral Potts transfer matrix. We show that the degeneracy of the eigenspace of τ 2 ( t q ) in the Q = 0 sector is 2 r , with r = ( N − 1) L / N when the size of the transfer matrix L is a multiple of N . We introduce chiral Potts model operators, different from the more commonly used generators of quantum group . From these we can form the generators of a loop algebra . For this algebra, we then use the roots of the Drinfeld polynomial to give new explicit expressions for the generators representing the loop algebra as the direct sum of r copies of the simple algebra .
We study the eigenvector problem in homogeneous superintegrable
N-state chiral Potts model (CPM) by the symmetry principal. Using duality symmetry and (spin-)inversion in CPM, together with Onsager-algebra symmetry and
sl2-loop-algebra symmetry of the superintegrable
τ(2)-model, we construct the complete
k′-dependent CPM-eigenvectors in the local spin basis for an arbitrary Onsager sector. In this paper, we present the complete classification of quantum numbers of superintegrable
τ(2)-model. Accordingly, there are four types of sectors. The relationships among Onsager sectors under duality and inversion, together with their Bethe roots and CPM-eigenvectors, are explicitly found. Using algebraic-Bethe-ansatz techniques and duality of CPM, we construct the Bethe states and the Fabricius-McCoy currents of the superintegrable
τ(2)-model through its equivalent spin-
2N−1-XXZ chain. The
τ(2)-eigenvectors in a sector are derived from the Bethe state and the
sl2-product structure determined by the Fabricius-McCoy current of the sector. From those
τ(2)-eigenvectors, the
k′-dependence of CPM state vectors in local-spin-basis form is obtained by the Onsager-algebra symmetry of the superintegrable chiral Potts quantum chain.
We demonstrate that the transfer matrix of the inhomogeneous N -state chiral Potts model with two vertical superintegrable rapidities serves as the Q operator of the XXZ chain model for a cyclic representation of with N th root-of-unity and representation parameter for odd N . The symmetry problem of the XXZ chain with a general cyclic representation is mapped onto the problem of studying the Q operator of some special one-parameter family of generalized τ (2) models. In particular, the spin-( N −1)/2 XXZ chain model with and the homogeneous N -state chiral Potts model at a specific superintegrable point are unified as one physical theory. By Baxter's method, developed for producing a Q 72 operator of the root-of-unity eight-vertex model, we construct the Q R , Q L and Q operators of a superintegrable τ (2) model, then identify them with transfer matrices of the N -state chiral Potts model for a positive integer N . We thus obtain a new method of producing the superintegrable N -state chiral Potts transfer matrix from the τ (2) model by constructing its Q operator.
We adapt our previous results for the “partition function ” of the superintegrable chiral Potts model with open boundaries to obtain the corresponding matrix elements of e −αH, where H is the associated hamiltonian. The spontaneous magnetization Mr can be expressed in terms of particular matrix elements of e −αH S r 1 e−βH, where S1 is a diagonal matrix. We present a conjecture for these matrix elements as an m by m determinant, where m is proportional to the width of the lattice. The author has previously derived the spontaneous magnetization of the chiral Potts model by analytic means, but hopes that this work will facilitate a more algebraic derivation, similar to that of Yang for the Ising model. KEY WORDS: Statistical mechanics, lattice models, transfer matrices. 1